NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL MEMORANDUM
No. 1114
THE CALCULATION OF COMPRESSIBLE FLOWS WITH LOCAL
REGIONS OF SUPERSONIC VELOCITY
By B. Gothert and K. H. Kawalki
Translation
Berechnimg kompressibler Stromungen mit
brtlichen tjberschallfeldern
Forsch-ungsbericht Nr. 1794
'^glE^
1 - Washington
I
\ March 1947
3 1176 014412259
"SAnmasiJiL adtjecuri caisxsnuEE for jqsrgkautics
TBCHErXCAL MEMQBAinXJM NO. llllv
TEE CALCDLATIOCT 093* CCMPSSSSIELS FLOWS VIXH LOCAL
HBGrXORS 0? SUFEBSQinJC vsLocrn^
By B. Gothart and. K, E. EamUd
ABSTRACT: Xbe follovlng report lo ooxLcenxod vLth a metbod for
the approxlmato oalculatlon of comproasl'ble flows
abont pznflloB vlth locel regions of supersonic
velocity. Tlw flow oi'ounfl. a slender profile Is
treated as oil exampll.e.
OIMJHE! I. Startenent of the PrpWLam.
H. Stirvey of the I^i^thod used.
III. Colculatloaa of the Bieurrple.
17. Appro3dinate Treaiaiont of Local BpglonB of Supernonlo
Telocity.
V, Synanoti'loal and Unarnsnetrlcal Koslone of Buper sonic
Flow.
VI. Summary.
I. STATSIEWP 0? THE PROBEE-I
Sevei<al methods ore Tmovn tor the calculation of ccmpresslhle
flovs at hl£^ subBonlc velocities. The resulting approximate
BolutlonB are quite -osefol as long aa sound velocity is not
exoeeded at any point of the flov field. Hovever, apparently all
these approximate cELLoulations, t/lthout exception, cease to
converge or to render useful flov pattozna If the condition of
purely suhsonlc flov Is no longer satisfied. Moreover, numerous
tests In vlnd tunnels conf limed the result that the reconversion
*"Berechnung JcdnpresBlhler StrflBrunson mit b'rtllchen Uber-
echallfeldem." Zentrale f(£r vlssenechaftllches Berlchtswesen
der Luftfahrtforschung des Generalluftzeugmslsters {2XIB) Berlln-
Adlerahof , Forechungaherlcht JTr. 179'^> Berlln-Adlershof, den
7. August 19'+3.
DACA IK So. llll^
of local supareoQlo flovB Into subsonic flovs la real flovs
takSB place not steeuUlT- Imt generally "by means of a oocoipreSBlon
slioclc which completely cbenf^s the vdiole flov pattezn^^ There-
fore, the queetion eriees Aether a contlnuouB process from
supersonic to subsoolo flov Is at all possllxLe for "bodies In
parallel flow, even thoue^i the foxmatlon of a "bovasdory layer on
the surface of the "body Is at first neQlected.
An appyoxlBifltlon me-Qiod vas tested In the JfTL In order to
olaarlfy these questions; this nusthod maJces possible the calculation
of flows with local regions of supersonic velocity.
II. SURVEY OV TBS MBraor TEED
The devolopnent of the method started frcm the fact' that the
Vnovn approximation msthodo for ccmpreselhls flows, tdilch are
without oiooptlon "based oa a step-by- step Improvecent of tlie
Inccoipresslble flov, are quite appropriate for tho llTolted liomaln
of the pure suboonlo flov eztendlnf; Srcsa. iJiflnlty to the sonic
voloolty boundary In the flov field near tiie body. Eovever,
for the region of the local supersonic ilov a method based on
the pxt^pertles of s'lpersonlc flov will be subsequently used, as
for Instonco the method of characteristics of Prandtl-Busemazin.
Accordingly the partial areas of supersonic and subsonic flow,
respectively, are calculr.ted SAparately by different methods
\dilch In each case arc treated according to the peculiarities of
the partial flow to be calculated. The partial flows that wore
thus detezmlned must then 'bo Joined in such a manner that the
flows agree on the surface of contact of the two regions, that is,
on tho sonic volocitj" boundary, with respect to magnitude as well
as to direction of the velocity. Figure 1 shove a scheoamtic
representation of the boundary between the two flov areas.
1. Subsonic Region with analT Perturbation 7eloolties
Frandtl's rule will repi-esent a good approximation for a
great part of the outer subsonic flov area, eijooe the perturbation
velocities caused by the profile are sufficiently small up to
Bcaae distance from the sonic boundary so that the assunrptiona of
no account will be taken here of flows -idilch ali^tlj' exceed
sound velocity because for them there is no certain way of
distingui^lng between a steady trsnsJtlon and a cccrpresslon
shock; neither by pressure -distribution measureoaents nor by optical
observations, for Instance according to the schlleran method. .
KAGA mso, mk
FraDcLtl's approximation are rather veil aatlsfled. The Bubsoolc
flov of this regloa-can, of cotErse, not "be o'btalned "by Blnsply
dlstorttog the Inoampresslhle flov around the profile aocordlng
to PrQiadtl*B rule. First, the local siiper sonic velocity field
requires essentially more spaoe than the suhsonlc flov. The
outer subsonic streamlines, therefore, ore vldened outvazrL not
only "by the hody In the flov tut also hy the additional displace-
ment due to the super sonla reloclty field. "Eoverer, this additional
vldenlng of the streazallnoa ly the aupersonlo Telocity field can
only affect the subsonic flov like a modification of the Isoundary
cooodltlons at the sonic Telocity houndary; such a modification
may he represented In a single way hy source end sink distrlhutlons,
dlpoles, end so forth.
2. Suhsonlo Eoglon vith El^ Porturhrtion Telocltles
As mantloDod "before, the pMrt of the siibsonlc flow vfaich has
t^ Tue calculated according to Pi'andtl does net extend as foi^ as
the sonic houndary; It only reachos up to a hounlrtry lino near
the sonic houndory determined "by an acreod siifflclontly small
Talue of the perturhatlcn Tol^oltles. (Tlila houndorj'- line is
represented In fig. 1 hy a datfied lino.) TIio calciilation for
the rejjlon trcm this line to the sonic Telocity 'bouadary must
generally "be csLLTlod out hy an ImproTed suhscnic method. A
nunorlcal method of calculation seems to he particularly appro-
prtate for this Inteimndiate region. The oiitor flow according
to Prandtl, which io assumed as known, nujy "by tiio mothod hy
continued a little further, namely up to the sonic Telocity
"boundary.
One assumes, for Instance, that the flow field is coTered "by
a rectangular grid of selected points. The Telocity cor^ononts
T^: and Ty at these points in the outer field are knotm for
each case according to. Prandtl* s approximation. Then the
Telocltles at the inner grid points may he calculated fixim the
Imown Taluss at the outer grid points if the differential
quotients vlilch are declslTe for this continuation are appro-
ximated "by the corresponding difference quotients.^ Figure 2
shall "be considered as an ex&mple. Magnitude and direction of
the Telocltles at all grid points outside of the "boundeury lines
are assumed as known. The air density shell he known also.
Then the exact equations of the continuity and the Irrotationality
are, for the indicated grid point "2"!
21 wish to express my thanks here to Dr. H. Schu"bert/D7L
for his suggestion to treat ccmpresslhle flc/'s according to
the difference method.
HAGA m No. llll^
>
or, In the notation of difference giiotlentEi!
•^1
^^3 ■ ^^1 Va ■ V ■ '
J
B-
If one asBranes the grid to "be qijadratlc vlth neahes of equal vldth
then ^ a ZSy and the equatlone gl7en aliove are eliapllf led as
follows:
'X2 ^ ^A,k * P2 ^ ^^r.A,k ■ -V2 ^ ^^,1 - P2 ^ ^^73,1 (1*)
"^^A,!. - ^^X3,i (It)
Frcm the second eqiiatlon one Immediately obtained the Telocity
component y-.. at the point A, -while the first equation gives
a relation hetvean the velocity ccairponont v and the air
density at the point A. Velocity and olr density are connected
In a rather ccanplicated way hy the adlahatlc relation
Pc
-^H'<@f
(p heire represents the eJLr density In the gas at rest a the
critical Bo-imd velocity) . ITherefore, It will he praotloal to
use the linearized relation Ap ■ j^ Av instead of the esaot
ftmctlon, the error "beinc nogllgihle, heoanse of the snuQl width
of the meshes:
HACA TM So. mh
l(p/Po) _-2,^. /pNi
— — — X --^ X
+ 1 ^
-^(#*-&)
Ccmpare figure 3.
With dT n -* X dT_ + -*-x dv" there results
T It ■ y
ra.
and "by siibatltutlon
^(p/Pq)
X At-. , + -Z£.x ^,
3CA,3t
^A,kj
At.
Va-^
^Ajlt
(^)
. T ▼ "~:
O + D ■ ■■ ■"—
d(p/pj
d^v/i
(3)
The Telocltj coiriponentB t
a-^jo
ani T.^. for the new grid pclnt
■ i/A
ero known, according to those ccLLciJ-atlona, frcm ecLne.tlone (1 "b) and (3).
Therefore- the air density also Is Imown accordlnf! to eqitatlon (2).
Thus the Intezsiedlate region between the- subsonic flow with nr.wn
perturlsatlon Telocltlos (calculation accordlnii to Premltl) and
the 'boundaxy may Tae detexmlned end the position cf the sonic
Telocity Ixjimdory and the Telocity direction alone that "boundary
will Tie o^btalnod as final result. It will haTe to "bo further
Investigated whether a calculation of the wliole s-jper sonic velocity
field Tieyond the eonlc houndary up to the surface of the Txidy "by
this method would "be practical. I!a the main, the time spent on
the calculation H and the accuracy of the method which has "been
developed here as contrasted with the metAiod of oliaracterlstlcs
of Prandtl-BuBQBiaiin will haTe to "be considered.
The difference method ^Aiich has "been cLeTelo?ed here does not,
howerer, pexnlt a start of the numerical calculation at an arlaltrazy
dlstanoe frcm the liody In the flow; for at a very great distance
frcan the "body the calculation would not be euff lolently accurate
since the moshes cf the grid v:<uld 1)0 too small. This connluslan
can bo d.^avn from the foot tliat at a gireat distance the flow about
any body without external forces can be replaced, except for "f"*^''''
ITACA mNo. ink
deviations, "bgr the flov field around a single dlpole. It Is
precisely tlie small derlatlons from the dlpole flov, lAiich exist
even at the greatest distance, that pezmlt a continuation of the
flov, for Instance according to the difference method, not a1x>ut
a cylinder "but around the special profile.
3. ^tgpersonlo Beglon
I^ram the calculations descrllied &Taove, vhlch were carried
out In the subsonic area according to Pran'ttl's rule, or hy the
difference method, the hoiaidary line of the supersonic velocity
field vas obtained. The Telocity along -Uils lino equals the
sound Telocity and the dlr-ectlom ef the flov at every point of
this line Is Imovn. The next problem consists In Joining the
corresponding sitpersonlc velocity field to this boundary line;
for Instance, by the method of oharewterl sties of Prandtl-
Busemann. According to the method of characteristics it is
Imovn that the changes of state occurring In a supersonic flov
are manifested In e^cpanslon cr ccnpresslon vaves. Therefore,
■vhen the flov passes throu^ thdtse lines of disturbance, the
flov velocity and the flov diretttloii are modified by definite
values 'Which can easily be detemlned from the graphical
representation of the chai-acteristicsi® It can be Inferred fjxm
the condition of constant velocity at the sonic boundaz? \ipjo
that tvD vaves must originate at every point of the siExzU'c'^undary,
namely an expansion and a compression vave. (See fig, h.) -Pie
velocity direction Junips from one starting point of these waves
at the sonic boundary to the next by a fixed amount for each case.
Since these starting points of the vaves can be shifted in snj
vay along the sonic velocity boundary, the beginning of a
graphical representation of characteristics may be dravn for
any given distribution of direction on the sonic boundary.
Lllcevlse the starting points of the vaves may be shifted not
only on the sonic boundary but also toirards points outside of
the sonic boundary, so that the graphical representation of
chai'acteristlcs nay be adjusted to any shape of the sonic
boundary line. TherevLth it has been demonstrated that at least
in certain cases there exists a possibility of continuing the
subsonic flov Into a corresponding supersonic flov; finally there
results the nev profile contour of the body in the flov/-
^Compare L. Prandtl: "Fuhrer durch die StromungBlehre . "
Verlag Vloveg and Sohn, Braunschvelg (19^2) S. 257 und folgende.
*In order to render possible sufficiently accurate dravlngs
of grids of characteristics for local supersonic velocity f leldjS,
a graphical representation of charactoristics vith an interval
of 1/5° has been completed at the STL; copies may be obtained.
BACA OMiro. ink
in. cALcncjfflioir op ah siamfee for a mixed
SDEEaasoNic-saBigoiiic itxw
A Blmple exsoDople will demanstrate the coAublnsd effect of the
calculation method^ for various flow reeXariB, TOie velocity field
for a two-dJmeiislanal Blonder "body vae determined In the proximity
of the Ixjdy "by means of comformfll trwn af oimatlon and tlie associated
distortion according to Prandtl for the Maoh number M » 0.86.-
Thus -Uiere resulted a sonic velocity "boundary as Indicated In.
figure 2(a). This sonic 'boundary, iftxloh vas ohtalned according
to Prandtl*a method, could "be further litiproved "by continuing the
flow step "by step to th^ Improved sonic velocity Twundnry; one
would have to start from a "boundary lino with sufficiently antfin
perturhatlon velocltios iistng the difference method descrihed
ahove. Thio corrfictlon of the sonic velocity "boundary vaa at
first diareiarded for the sako of simplicity, Elnco in c©neral,
it can "bo noglectod for tho "basic calculatlona pltnned for this
report. By means of the directions of velocity alone "^e sonic
"boundary, the neb f,f characteristics for the Eiuporsonic flow can
"be drawn so that the contour of the "body in the flow la a streami-
line of this siiporscnic field. Only expansion waves start from
the Bvirface of the "body, t&iile cantpresslon vaves, Tiilch are
necessecry for a roconversion of the supersonic flew into a sub-
sonic flow start witliout exception from the sonic "boundary. This
"behaviour la an escontial criterion of the grid of characteristics.
The manner of reconversion of the supersonic into a subsonlo flow
(that is, for instance, \diether ateadlly or "by means of a com-
pression shock) is doclsively Influenced "by the sonic vclocltj'-
"boundary.
Of course, the contoiTT of the "body at the "boundary of the
Bizpersonic field idiich has "been determined in the foregoing
calculation of the example can not coincide with the contour of
the original "body idilch was originally distorted according to
Frandtl, since the supersonic field, "because of the expansion
of the air, requires more space than the flow field ^ich was
distorted according to Prandtl. Therefore the surface of the
"body in the supersonic region will "be flatter than the surface
of the original hody in order to satisfy this Incroeised need for
space. The oixter su'bsonlc flow must "be adjusted to this increased
need of space of the supersonic field In order to o"bteln -Qie
original contour as the result of the calciilatian. This adjustment
may "be made "by replacing the supersonic field with dlpoles or
source-sin]): distrl"butlons, tdilch will widen tho stroamlines of
the su"bsc(Qlc region originally ohtained "by Frandtl* s method. The
strength of this source-sink distrl'butlon is Imown fj^Gsn the
oc(ndltlcn that the space idilch is to "bo added has to cover tho
8 IfACA. IMNo. 1114
difference of the flov density In eupersonlo flov and flov
according to Prandtl and that this space corresponds In
magnitude to the flattening of the original "body *hlch resulted
in figure 5(a). Bj- a stop-hy-step approximation the contour
idilch results at the end of the culculatlon loay thus he
adjusted to the desired ocntotcr.
Hovever, the eTZonple treated here already permltc the
ccncluplon that vbtxa. the local sound \elocity Is exceeded there
exist solutions vlth reccnTorslcaa of the local s'aper sonic
velocities into suhsonlc velocltloB vlthout conpi'eBslon shock,
at least In frlctloolees flow.
The pressure dlstrihiitlon >ailch wp.s fo'ind for the example
treated.- Is congjared to the prensure 'iJ.sta.'l'bu.ttf-n of tnother
slender "body (fig. 5(hJ tixldh was deteminod acc^i'dlnc oo the
same lav of confozmal mapping and shculd a^^reo voll td-th the
hody of the example. (Soe fig. 5(&).) ^^3 extriiune flitnesa
of the pi-esQure dlstrlhutlon In the svpersorlc region in a
special chai-acterlBtlc for the hody with local reclon of super-
sonic velocity. This hehaviour coiTespmnds to tho sti-ong
curvati-re of the contour of the "body imraodlately after the
flov has ontered the supersonic region. D-jo to this large
lorjal curvature, the hody 3 In this region vill produce even
■for InccsqpresBible flov higher nege.tlve propsurea thpn the
hody A. For this reason the two pressure dJ.strlbuticns Tdilch
■♦•ere graphically rrprosentod cpnnot "no 'lireotlj ccoiparod.
IV. APEROECMAEE TREMMEHT OP LOC^L EBGIOHS
OF SUSBE30KIO VELOCITY
The representations of characteristics considered thus far
vers notable for the fact that expansion or ccanpresslon vaves
running in the same direction never intersect. (Ccsoiparo fig. 5(a).)
An Intersecting of vaves running In the same diroctli..n is basically
Impossible for expansion vaves since such 'VTUves 'jLLvoys diverge
from their starting point. But canrpresalon wives oonvjrge; there-
fort^ they could veil foim an envelope and cause a compression
shoolc.
Such overlapplngs of vaves vere found for the sample body of
figure 5(a) '^dien the free-strenm velocity vae inci^eased from
M B 0.86 to H a 0.90. The ccnotructlon of the oxpaasian vaves
vas started at the sonic boundary (In opposition to their actual
direction) and already for the Mach number M a 0.86 (according
TIACA TMNo. 232.k
to figure ^(a)) led to a pressing together of the expansloti
vaves near" the contoizfc'; for* the Mach ninaber M'b 0^90 - they
ran Into each other even hefor^ reaching; the oontotir. ^Is
condltlan Is physically senseless, that Is, a sjaanetrlcal
solution for the sonic houndary iftilch iris found 'according to
the Indicated scheme Is, at this Maoh numher M ■ 0^90, no
longer posslhle. In this case the "expansion shock,'.' according
to a consideration of Frandtl, rather will dissolve Into a
group of divergent expansion waves orlg^jiatlng at the contour,
vhlle the compression shuck idilch liialts the Erupersonlc area
persists. Ill this manner one easily phtalns the flov picture
found from many tosts for.'nblch the local sitpersonlc flov Is
no longer reconveirted Into the su'bsonlc flov throu^ a contlnfjous
phenomenon hut hy means of a cosapresslon shock.
The preceding treatment vas hosed upon tho sonic velocity
"boundary line idilch vas ohtolned according to Prandbl's rule
without additional sou2'ce-3lnk todies wilch woiild have corre-
sponded to the groater neei for space of the Buperaaalc reclon.
The reeultlng solution tflilch was physically senselesH mi^t
veil he caused hy the perhaps no longer appropriate conditions
for the exlsteonce of the sonic velocity houndaay line. In order
to ohteln a cleeirer vlev one consldei^s the llialtlng case ^ere
thT expansion waves converge exactly at the contoirr. A flov
around a coxner develops, as represented schemctlcally In
flgirre 6; In tho s'.Tpersanlc reclon such a flow Is pooslble
vlthout flow separations. Tbin aosunptlon and a given sonic
velocity hotmdary offer the supersonic region tho opportizilty
to fill the largest posslhle space without concave curvature of
the hody In the Bupersonld domain.
An admission of concave surfaces would presuppose a sonic
velocity houndaiy ■which could no longer he produced from an
Incompresslhle flov ahout a profile vlthout additional source
and sink dlstrlhutlons In place of the supersonic region; or
else one would have to drop the condition of f].ov sjonmetry.
?lgio7e ^(a) demonstrate 8 clearly hov the surface always shows
a convex curvatiUTs when the inccanlng wave Is a comprosslon wave,
the outgoing wave an expansion wave. However, for the opposite
condition of concave curvature In symmetrical flov tlie ccmlng
In. of an expansion wave and going out of a compression w&ve
would iresult In the amission of expansion waves from the sonic
boundary In hoth cases'. But an expansion wave vhlch stairts at
the sonic houndary cannot he obtained hy sl3iq;>ly exchanging the
cantpresslon end expansion waves, for Instance according to the
diagram In figure h; for there must always first appear an
expansion wave and thun a compression wave In tho direction of
the flov at the sonic velocity houndary, because the flov can
10 KACA OMSo. lull-
peisB flxaa the soolo tova&jeaj Into the sttperBonlc region onljr
ty means of an expansion, ko. expansion vave stajrtlne kt the
sonic botmdazy Is posslMe only for a certain shape of the
comers In the graphical representation of characteristics:
tiro expansion vaves are sent out from the comer BlTrmltwneov.aljr.
(Ccaapeace fig. 7>) There Is, hovever, a condition; the sonic
houndaiy In this region must talce a steeper course than the
expansion vave vhlch Is Inclined tovard the direction of the
flow 'by the Maoh angle. The strecmline through snich a comer
point lies hetween the expansion -waves, 'Hhlle for a noimal comer
It Is tangent to the apex of the triangle foimed liy the expansion
and compresBloa -ware.
It la an Important criterion for the comer vLth tvo expansion
vsrea that the streamlines heoome steeper, not flatter, vLth
Increasing distance from the tody; othei*wlse the flov -Hould not
fit together after transition throu^t the two expansion wavas.
This reemlt agrees vlth the fact that In a B7mr<jotrlcal supersonic
region also the direction of the atreemllnes grows steeper with
Increasing distance from the profile. The catise of this phencanenon
Is that the stroaaillnes (hecause of the mflTHmnw flov density at
Bonlo veloolty) are closest to each other at the sonic boundary
vhlle there vlll he the greatest distance hetvoen them for the
Btipersonlc region at the point of Tna-rlTmam velocity, that Is,
generally, at the jKslnt of greatest thldmess of the hody.
Such dletrlhutlons of slopes occur only In supersonic regions;
they are not posslhle in suhsonlc areas for a flov ahout hodles
vlth convex contotirs. Therefore it la not surprlslnn that the
sonic velocity hoxmdary, \dilch vas obtained by distortion of the
Incompressible flov about the body, vlll not load to a useful
supersonic flov pattern as long as there are no singularities,
as for instance source- slnlc distributions in the incompressible
flov by means of lAiich the effect of the increased, expansion of
the air in the supersonic region can be calculated.
Presently a method is under investifjatlon in -vhich th? sonic
velocity boundary may be adjusted to tho increasotl need for
space of the supersonic flov by arrangement of singularities in
the flov; the results of these investigations vlll soon be
published separately.
Y. SQMSERICAL AFTO msmmsOCAL SBQIORS OF SURERSOHIC FLOV
As stated in detail in the preceding chapter, concave body
suz^'aces in the supersonic region are dependent t^pon the porticular
BACA ON No. lllk 11
type of comers In the net of ohaxaoterlstlcB -Nhlch le characterlEed
"by tvo ezpansloD, vavee starting frcoa a point of ^ the soolc 'bottDdazT'.
(Canipa3?e fig. 7.) The stlpulatian of flov STmMtry then causes
compresslan waves to start from the corresponding points of the
contour tdilch In turn cause these cnrnpresslon waves to proceed
toward the sonic houndarj In pairs . Since these ccdZQresslon vaves
'vbich Dtaiii at the contour alvays converge, such solutions ere
posslhle only as long as the radiated coaiipresslaa vaves do not
Intersect; othorvlBe a oonipresBlon shock vlll develop. Therefore
the existence of STnanatrlcal flovs vlth local excpersonlc regions
cannot "be counted upon for strongly concave curvatinres or for hlgjh
Mach numbers, "because the ccsapresslon shoclc -would presuppose a
symmetrical expansion shoclc In a corre°spondlng locatloa; li^ls
ea^panslon shoclc, hovever, la not physically possllile.
ISxcept for the special case descrl'bed aTx>ve, st^ersonlo flows
on principle tend toward flow e;mnistry as will he shown In the
exanrple represented In figure 8(a). Thore vlll alTrays develop a
symmeti-lcal flow alang the wall with two chances In direction for
IncQznpressl'ble flow without separations. It Is, however, lmo>qa
that for pure supersonic flow the flow along the fTall Is unsynmetrlcaL
(fig. 8(a)), If no additional guiding surfaces exist. Conditions
of BTnmetry can "be achieved throu^^i calculations for the mixed
BuTjoonlc- supersonic flow around this douhle comor: the outer
subsonic flow at the sonic houndary will supply the missing
synmetrlcal "boundaiy conditions (as for the exEmple In fig, 5(a)).
But It does not appear Imposslhle that such a mixed superscnlc-
BulsBonlc flow fdilch has "been made symmetrical "by the outer subsonic
flov may he unstahle. Since the suhacnlc flov Is produced "by the
form of the ohstacle In the' flow It t-jIII "bo symmetrical only for
a sjonmetrlcal ohstacle. On the other hand the cuter flow will not
"be Bynmietzlcal for an unsymmetrlcal obstacle, as for Instance a
symmetrical "body with a local supersonic region and a can^tresslon
ehock. Therefor^ It seems very doubtful whether the outer flow
vbose fozra la decisively determined "by the foxm of the supersonic
region, will In turn "be able to roEihape decisively the supersonic
flov by forcing the symmetry conditions upon It.^
A oonf Iznatlon of the noneymnstry of the symnetrloal type of
solution would explain the occurrence of unsjianetrlcal solutions
with ccoDipresslon shoclcs idilch has been observed In tests.
On this, occasion X should lUce to point otct that a ajmnetrlcal
supersonic flov also could be suggested for the single wall represented
^A solution of this stability problem shall be obtained by
assuming a fgnan unsynnetrlcal defoznatlon of the sonic boundary and
by then observing fAiether this defoznatlon Increases or dlmlnlElhes.
12 m.ck m iTo. 1114
in flsare 8(a) idilch Is ^bTsloally not real "but satisfies the
conditions for potential flows vltb regard to contlntcLtj and
Irrotatlonall ty as veil as Beznoulll's equation. This flotr Is
represented jji figure SCb). Such a flow, aa Is vbU Imovn, Is
phyBlcally real for the reason that a dlsturljance can not "be
tranamltted upstream In supersonic flov. For solutions vhlch
vero o1}talned gnalytlcftlly It must, therefore, alvays "be oFipeclally
verified ^etber this additional condition has heen satisfied for
the supersonic florf.
71. SafMHT
1. A method of calculating the approximate Telocity field
for cortpresBlhl© fi-ows vlth local regions of supersTnlc velocity
has heon presented. Starting from the flov at a lar^ distance
from the profile deteimlned' according to Franltl's rule, this
outer flov was continued t-o the sonic Telocity "boundary ty means
of a numerical methcd; the metiiod ct characteristics of Frandtl-
Busemann was applied for continuation "beyond that "boundary. These
calculations result finally In the contour of the profile In the
region of supersonic Telocity.
2. It has "been deinonstrated In an exazople that mixed subsonic-
supersonic flows a'bcut two-dimensional "bodlos can "be calculated
tdiere net only the transition fi-om su'bsonlc to supersonic "but also
the transition from. s''rpersonlc to su'bsonlc taJces place continuously,
that Is, "vTlthout pressure Jump.
3. ITo physically real local area of supersonic Telocl'ty could
"be determined for the example considered here -vihen the sound
Telocity was far exceeded "because then esqpanslon or scmpresslon
shocks occurred. Howrarer, there Is a prospect of calculating mixed
flovB for such cases also: "before starting the calculation one
would hare further to extend the oiiter su'bsonlc flow "by means of
source and sink distributions; In this way tJie greater need, for
sp£u;e of the supersonic Telocity field would "be satisfied.
l^. The symmetry of the local area of supersonic Telocity for
mixed supersonic- su'bsonlc flows may "be enforced "by the outer
su'bsonlc flow. Since, hoverer, the outside subsonic flow In
turn must "be produced "by the profile In the flow or "by "the local
area of supersonic Telocity, the assuioptlon seems Justified that
BAGfL OMNo. 111b 13
often only an Tgista'ble BQnraetrloal flov can "be pcroduced "by the
outolde subsonic flov; and this loistable Bjnmetrlcal flov vlll
turn even at email dlsturljances Into the unEywmotrloal case vlth
cceoopresBlon efhooks.
Translated "by M&ry L. Mahler
ITatlotoal Mrlsoxy Ccemnlttee
for Aoronautlos
>
Sonic velocity-Boundary
12!
o
Figure 1. Schematic representation of the flow regions for mixed subsonic
-supersonic flow.
Fig. 2
NACA TM No. 1114
Figure 2. Partition of the flow field for the difference method.
NACA TM No. 1114
Figs. 3,4
^Q
io
via.'' Z,6
Figure 3. Relation between air density and velocity under the assumption
of an adiabatic change of state.
Subsonic
re gi on
Sonic velocity-Boundary
Direction of velocity
Supersonic
region
Figure 4. Sonic boundary with the field of expansion and compression waves,
respectively.
Sonic velocity-Boundary
Expansion waves
Compression waves
Direction of the
oncoming flow
(-■•
CJI
P3
Direction jump in the net of characteristics
O
>
Figure 5a. Grid chacteristics in the local supersonic region for a two-dimensional body
B (relation of thickness d/1 = 0.0715 at the Mach number M = 0.86.
g
i4:>-
ipl,,
Profile
A: Pressu
at rfec
re distribution
i-number M=
_Jf^
.-*j —
-^
I
y
^
^-'"^
s.
^
N
\
14
IT ^
\ — \ — 1 — \ — r — r
0,2 H/ 1,1 0,1 1,0
o
>
o
>{^
ProfHeAfdll'Umi)
Promidli'-om )
Figure 5b. Calculated pressure distribution for two two-dimensional bodies in incompressible
flow and in compressible flow with local area of supersonic velocity.
Figs. 6,7
NACA TM No. 1114
Expansion waves
Sonic velocity-Boundary -^,/x
Compression waves
Direction of velocity
Figure 6. Mixed subsonic-supersonic flow about a body with plane surface in
the supersonic region.
Normal corners
Direction of the
oncoming flow
Sonic velocity-Boundary
Expansion wave
Compression wave
Figure 7. Schematic representation of the two different kinds of wave
radiation at the sonic velocity boundary.
NACA TM No. 1114
Fig. 8
Velocity- Vector
Compression
Figure 8a. Supersonic flow about a double comer.
Velocity-Vector
Expans i on
Expansion
^7777777777
Figure 8b. Symmetrical potential flow with supersonic free-stream
velocity about a physically not realizable double corner.
NASA Technical Ubrai
3 1
76 01441
2259